A particle starts from the origin at t=0 with a velocity of 8.0 ĵ+2.0 ĵ moves in the x y plane w

step 1 So for part A, given that the initial velocity vector, the initial equaling 8 .0 meters per seco

A particle starts from the origin at t=0 with a velocity of...

A particle starts from the origin at $t=0$ with a velocity of $8.0 \hat{\mathrm{j}}+2.0 \hat{\mathrm{j}}$ moves in the $x y$ plane with constant acceleration $(4.0 \hat{\mathrm{i}}+2.0 \hat{\mathrm{j}}) \mathrm{m} / \mathrm{s}^{2}$ . When the particle's $x$ coordinate is $29 \mathrm{m},$ what are its (a) $y$ coordinate and (b) speed?

A particle starts from the origin at $t=0$ with a velocity of $8.0 \hat{\mathrm{j}}+2.0 \hat{\mathrm{j}}$ moves in the $x y$ plane with constant acceleration
$(4.0 \hat{\mathrm{i}}+2.0 \hat{\mathrm{j}}) \mathrm{m} / \mathrm{s}^{2}$ . When the particle's $x$ coordinate is $29 \mathrm{m},$ what
are its (a) $y$ coordinate and (b) speed?

Key Concepts

Vector Kinematics

This concept involves the use of vectors to represent quantities having both magnitude and direction, such as displacement, velocity, and acceleration. In two-dimensional motion, each vector is expressed in its component form, allowing independent analysis in the x and y directions.

Uniformly Accelerated Motion

Uniform acceleration means that the acceleration remains constant throughout the motion. This allows the use of kinematics equations, which relate displacement, velocity, acceleration, and time for each component of the motion independently.

Component Analysis of Motion

By breaking the motion into its x and y components, the problem can be solved separately for each direction. This method leverages the fact that the motion in orthogonal directions is independent, making it easier to determine time and displacement along one axis and then applying that information to the other.

Time Parameter Elimination

This technique uses one component's kinematic equation to solve for the time variable, which is then substituted into the equation for the other component. It is a common strategy in problems where information is given for one coordinate and needed for another.

Magnitude of Velocity

After obtaining the velocity components in the x and y directions, the overall speed (or magnitude of the velocity vector) is found by applying the Pythagorean theorem. This combines the two perpendicular components into a single scalar quantity representing the rate of motion.

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